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I'm trying to solve a saddle point problem of the following form: Let $(E,\mathcal E,\lambda)$ be a measure space; $p$ be a probability density on $(E,\mathcal E,\lambda)$ and $\mu:=p\lambda$ $W$ be ...

TL;DR: I want to find a definition generalizing "$t \mapsto \frac{1}{m} \sum_{k = 1}^{m} \delta_{x_k(t)}$ is a Wasserstein gradient flow" to matrix-valued probability measures. Let $(X, d)$ ...

Suppose I have a functional of the form $$ F(\mathbb{P})\triangleq \int_{\mathbb{R}^d} \int_{\Omega}f(x,\omega)\mathbb{P}(d\omega)m(dx), $$ where $m$ is the Lebesgue measure and $\mathbb{P}$ is a ...